Editing Fractional-order control

{{Short description|Field of mathematical control theory}}
{{Refimprove|date=January 2009}}

'''Fractional-order control''' ('''FOC''') is a field of [[control theory]] that uses the [[fractional-order integrator]] as part of the [[control system]] design toolkit. The use of fractional calculus can improve and generalize well-established control methods and strategies.<ref>Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D. and Feliu-Batlle, V., 2010. Fractional-order systems and controls: fundamentals and applications. Springer Science & Business Media.https://www.springer.com/gp/book/9781849963343</ref>

The fundamental advantage of FOC is that the fractional-order integrator weights history using a function that decays with a [[power-law tail]]. The effect is that the effects of all time are computed for each iteration of the control algorithm. This creates a "distribution of time constants", the upshot of which is there is no particular time constant, or [[resonance frequency]], for the system.

In fact, the fractional integral operator <math>\frac{1}{s^{\lambda}}</math> is different from any integer-order rational [[transfer function]] <math> {G_{I}} (s)</math>, in the sense that it is a non-local operator that possesses an infinite memory and takes into account the whole history of its input signal.<ref>{{cite journal |first1=M.S. |last1=Tavazoei |first2=M. |last2=Haeri |first3=S. |last3=Bolouki |first4=M. |last4=Siami |title=Stability preservation analysis for frequency-based methods in numerical simulation of fractional-order systems |journal=SIAM Journal on Numerical Analysis |volume=47 |pages=321–338 |year=2008 |doi= 10.1137/080715949}}</ref>

Fractional-order control shows promise in many controlled environments that suffer from the classical problems of overshoot and resonance, as well as time diffuse applications such as [[thermal dissipation]] and chemical mixing. Fractional-order control has also been demonstrated to be capable of suppressing chaotic behaviors in mathematical models of, for example, muscular blood vessels<ref>{{cite journal|last1=Aghababa|first1=Mohammad Pourmahmood|last2=Borjkhani|first2=Mehdi|title=Chaotic fractional-order model for muscular blood vessel and its control via fractional control scheme|journal=Complexity|volume=20|issue=2|pages=37–46|doi=10.1002/cplx.21502|year=2014|bibcode=2014Cmplx..20b..37A}}</ref> and robotics.<ref>{{Cite journal |last1=Bingi |first1=Kishore |last2=Rajanarayan Prusty |first2=B. |last3=Pal Singh |first3=Abhaya |date=2023-01-10 |title=A Review on Fractional-Order Modelling and Control of Robotic Manipulators |journal=Fractal and Fractional |language=en |volume=7 |issue=1 |pages=77 |doi=10.3390/fractalfract7010077 |issn=2504-3110 |doi-access=free}}</ref>

Initiated from the 1980's by the Pr. Oustaloup's group, the CRONE approach{{huh?|date=November 2024}} is one of the most developed control-system design methodologies that uses fractional-order operator properties.{{CN|date=November 2024}}

==See also==
* [[Differintegral]]
* [[Fractional calculus]]
* [[Fractional-order system]]

==External links==

* [http://mechatronics.ucmerced.edu/research/applied-fractional-calculus Dr. YangQuan Chen's latest homepage for the applied fractional calculus (AFC)]
* [http://sites.google.com/site/fractionalcalculus/ Dr. YangQuan Chen's page about fractional calculus on Google Sites]

===References===

{{reflist}}

[[Category:Control theory]]
[[Category:Cybernetics]]


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